Hİ. How can we sure that the initial conditions , say, for a second-order linear equation must be unique which is also the uniqueness of the solution.
Yes, it is about uniqueness theorem from the initial conditions viewpoint.I think I just answered this in the thread "a property of differential equations" so I won't repeat all. Please read that.
However, I'm not sure what you mean by "the initial conditions must be unique". Are you simply referring to the uniqueness of the solution satisfying initial conditions.